---
title: "A student conducts an experiment to determine the unknown positive charge \\(q\\) on a small sphere, Sphere A. Sphere A has a mass \\(m = 1.0 \\text{ g}\\) and is suspended from the ceiling by a lightweight, nonconducting string of length \\(L\\). A second small sphere, Sphere B, carries a known positive charge \\(Q = +5.0 \\times 10^{-8} \\text{ C}\\) and is mounted on an insulating stand. Sphere B is brought near Sphere A so that their centers are at the same horizontal height, separated by a distance \\(r\\). The repulsive electric force causes Sphere A to deflect from the vertical by an angle \\(\\theta\\), as shown in Figure 1. The student assumes the spheres act as point charges."
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url: "https://nerd-notes.com/ubq/117843/"
date_modified: "2026-08-04T07:59:12+00:00"
---

# A student conducts an experiment to determine the unknown positive charge \(q\) on a small sphere, Sphere A. Sphere A has a mass \(m = 1.0 \text{ g}\) and is suspended from the ceiling by a lightweight, nonconducting string of length \(L\). A second small sphere, Sphere B, carries a known positive charge \(Q = +5.0 \times 10^{-8} \text{ C}\) and is mounted on an insulating stand. Sphere B is brought near Sphere A so that their centers are at the same horizontal height, separated by a distance \(r\). The repulsive electric force causes Sphere A to deflect from the vertical by an angle \(\theta\), as shown in Figure 1. The student assumes the spheres act as point charges.

A student conducts an experiment to determine the unknown positive charge \(q\) on a small sphere, Sphere A. Sphere A has a mass \(m = 1.0 \text{ g}\) and is suspended from the ceiling by a lightweight, nonconducting string of length \(L\). A second small sphere, Sphere B, carries a known positive charge \(Q = +5.0 \times 10^{-8} \text{ C}\) and is mounted on an insulating stand. Sphere B is brought near Sphere A so that their centers are at the same horizontal height, separated by a distance \(r\). The repulsive electric force causes Sphere A to deflect from the vertical by an angle \(\theta\), as shown in Figure 1. The student assumes the spheres act as point charges.

![A horizontal line representing a ceiling. A vertical dashed line extends downwards from the ceiling. A string is attached to the ceiling at the exact point where the dashed line begins, extending downwards and angled to the left. A small sphere labeled 'A' is attached at the end of the string. An angle arc labeled \(\theta\) is shown between the vertical dashed line and the string. A second small sphere labeled 'B' rests on top of a vertical stand, positioned to the right of the vertical dashed line. Both Sphere A and Sphere B are exactly at the same horizontal level. A horizontal double-headed arrow labeled \(r\) spans between the center of Sphere A and the center of Sphere B. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830351-uUal56.jpg)

**Part a)** **Derive** an expression for \(\tan\theta\) in terms of \(q\), \(Q\), \(r\), \(m\), and fundamental constants. Show all steps in your derivation. *(3 points)*

**Part b)** The student wishes to determine the charge \(q\) by collecting data for the angle \(\theta\) at various distances \(r\). **Indicate** which quantities could be graphed on the vertical and horizontal axes to yield a straight line whose slope could be used to calculate \(q\). Vertical Axis: _______________ Horizontal Axis: _______________ *(1 points)*

**Part c)** The student collects the data shown in the table below. | Trial | \(r \text{ (m)}\) | \(\theta \text{ (degrees)}\) | | | |-------|--------------------|-------------------------------|-|-| | 1     | 0.050              | 20                            | | | | 2     | 0.060              | 14                            | | | | 3     | 0.070              | 11                            | | | | 4     | 0.080              | 8                             | | | | 5     | 0.100              | 5                             | | | **Calculate** the values needed to graph the relationship you identified in part (b) and record them in the blank columns of the table above. Then, **plot** the data points on the axes provided and **draw** a best-fit straight line. *(4 points)*

**Part d)** Using the slope of your best-fit line, **calculate** the experimental value of the unknown charge \(q\). *(3 points)*

**Part e)** For the larger values of \(r\), the angle \(\theta\) is very small. **Explain** why measuring the horizontal displacement of Sphere A from the vertical dashed line using a ruler might lead to less experimental uncertainty than measuring the angle \(\theta\) directly with a standard protractor. *(1 points)*

**Part f)** A large, initially uncharged, grounded conducting plate is carefully lowered exactly halfway between Sphere A and Sphere B without touching either sphere. **Predict** the new equilibrium position of Sphere A. - [ ] Sphere A will remain deflected away from Sphere B. - [ ] Sphere A will hang perfectly vertically. - [ ] Sphere A will be deflected toward the conducting plate. **Justify** your answer. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117843/*